Sun's (Oracle's really) JVM, for the most part, yes, but it doesn't have to be. The language used to create the JVM is completely irrelevant as long as it is able to execute the byte code in the class files in the proscribed manner. And the compiler is written in Java, BTW, and uses the JVM.
Edit: P.S., when using an IDE you are, normally, not using the compiler provided by the JDK (whether Oracle's, IBM's, or the OpenSource Community's, or any other that may exist), you are, then, using the compiler that the IDE provides.